### Special Math Colloquium

# Alexey Kanel-Belov

### (Bar-Ilan University)

## "Interlocking structures"

** Date: ** |
December 20, 2021 |

** Time: ** |
15:30 |

** Place: ** |
tba (Jacobs University) and online via zoom (please write s.petrat AT jacobs-university.de for the meeting id and passcode) |

**Abstract:** Consider a set of contacting convex figures in $R^2$. It can be proven that one of these figures can be moved out of the set by translation without disturbing others. Therefore, any set of planar figures can be disassembled by moving all figures one by one. However, attempts to generalize it to $R^3$ have been unsuccessful and finely quite unexpectedly of convex bodies were found. Author proposed a follows mechanical use of this effect. In a small grain there is no room for cracks, and crack propagation should be arrested on the boundary of the grain. On the other hand, grains keep each other. So it is possible to get "materials without crack propagation" and get new use of sparse materials, say ceramics. Quite unexpectedly, such structures can be assembled with any type of platonic polyhedra, and they have a geometric beauty. Some patents were obtained https://www.elibrary.ru/item.asp?id=47260049, https://www.elibrary.ru/item.asp?id=47259870, https://www.elibrary.ru/item.asp?id=46607120 The talk is devoted to the different structures. The talk is devoted to the theory of self-interlocking structures and to the recent progress in it by Manturov: a) There exist two-dimensional self-interlocking structures in 3-dimensional space; b) One can construct self-interlocking 2-dimensional structures which are rigid once two polygons are fixed. Vassily O. Manturov, Alexei Kanel-Belov, Seongjeong Kim, Two-dimensional self-interlocking structures in three-space, 2021 (Published online) , 21 pp., arXiv: 2109.06426. Kanel-Belov, A.J., A.V. Dyskin, Y. Estrin, E. Pasternak and I.A.Ivanov. 2010. Interlocking of convex polyhedra: towards a geometric theoryof fragmented solids. Moscow Mathematical Journal, arXiv:0812.5089v1. Dyskin, A.V., Y.Estrin, A.J.Kanel–Belov and E.Pasternak,“Interlocking properties of buckyballs.”, Physics Letters A, 319 (2003),373–378 jumas, L., Simon, G.P., Estrin, Y. et al. Deformation mechanicsof non-planar topologically interlocked assemblies with structuralhierarchy and varying geometry. Naure, Sci Rep 7, 11844 (2017).https://doi.org/10.1038/s41598-017-12147-3